Properties

Label 900.1.bh.a
Level $900$
Weight $1$
Character orbit 900.bh
Analytic conductor $0.449$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 900.bh (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.449158511370\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{40}^{11} q^{2} -\zeta_{40}^{2} q^{4} + \zeta_{40}^{3} q^{5} + \zeta_{40}^{13} q^{8} +O(q^{10})\) \( q -\zeta_{40}^{11} q^{2} -\zeta_{40}^{2} q^{4} + \zeta_{40}^{3} q^{5} + \zeta_{40}^{13} q^{8} -\zeta_{40}^{14} q^{10} + ( \zeta_{40}^{2} - \zeta_{40}^{16} ) q^{13} + \zeta_{40}^{4} q^{16} + ( -\zeta_{40}^{9} + \zeta_{40}^{17} ) q^{17} -\zeta_{40}^{5} q^{20} + \zeta_{40}^{6} q^{25} + ( -\zeta_{40}^{7} - \zeta_{40}^{13} ) q^{26} + ( \zeta_{40} - \zeta_{40}^{3} ) q^{29} -\zeta_{40}^{15} q^{32} + ( -1 + \zeta_{40}^{8} ) q^{34} + ( -\zeta_{40}^{8} - \zeta_{40}^{10} ) q^{37} + \zeta_{40}^{16} q^{40} + ( -\zeta_{40} + \zeta_{40}^{7} ) q^{41} + \zeta_{40}^{10} q^{49} -\zeta_{40}^{17} q^{50} + ( -\zeta_{40}^{4} + \zeta_{40}^{18} ) q^{52} + ( \zeta_{40}^{15} + \zeta_{40}^{19} ) q^{53} + ( -\zeta_{40}^{12} + \zeta_{40}^{14} ) q^{58} + ( \zeta_{40}^{14} + \zeta_{40}^{18} ) q^{61} -\zeta_{40}^{6} q^{64} + ( \zeta_{40}^{5} - \zeta_{40}^{19} ) q^{65} + ( \zeta_{40}^{11} - \zeta_{40}^{19} ) q^{68} + ( -\zeta_{40}^{4} - \zeta_{40}^{18} ) q^{73} + ( -\zeta_{40} + \zeta_{40}^{19} ) q^{74} + \zeta_{40}^{7} q^{80} + ( \zeta_{40}^{12} - \zeta_{40}^{18} ) q^{82} + ( -1 - \zeta_{40}^{12} ) q^{85} + ( -\zeta_{40}^{15} - \zeta_{40}^{17} ) q^{89} + ( -\zeta_{40}^{6} + \zeta_{40}^{8} ) q^{97} + \zeta_{40} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 4q^{13} + 4q^{16} - 20q^{34} + 4q^{37} - 4q^{40} - 4q^{52} - 4q^{58} - 4q^{73} + 4q^{82} - 20q^{85} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{40}^{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−0.156434 0.987688i
0.156434 + 0.987688i
−0.156434 + 0.987688i
0.156434 0.987688i
0.453990 0.891007i
−0.453990 + 0.891007i
0.453990 + 0.891007i
−0.453990 0.891007i
0.891007 0.453990i
−0.891007 + 0.453990i
0.891007 + 0.453990i
−0.891007 0.453990i
−0.987688 0.156434i
0.987688 + 0.156434i
−0.987688 + 0.156434i
0.987688 0.156434i
−0.987688 + 0.156434i 0 0.951057 0.309017i 0.453990 + 0.891007i 0 0 −0.891007 + 0.453990i 0 −0.587785 0.809017i
287.2 0.987688 0.156434i 0 0.951057 0.309017i −0.453990 0.891007i 0 0 0.891007 0.453990i 0 −0.587785 0.809017i
323.1 −0.987688 0.156434i 0 0.951057 + 0.309017i 0.453990 0.891007i 0 0 −0.891007 0.453990i 0 −0.587785 + 0.809017i
323.2 0.987688 + 0.156434i 0 0.951057 + 0.309017i −0.453990 + 0.891007i 0 0 0.891007 + 0.453990i 0 −0.587785 + 0.809017i
467.1 −0.891007 0.453990i 0 0.587785 + 0.809017i −0.987688 + 0.156434i 0 0 −0.156434 0.987688i 0 0.951057 + 0.309017i
467.2 0.891007 + 0.453990i 0 0.587785 + 0.809017i 0.987688 0.156434i 0 0 0.156434 + 0.987688i 0 0.951057 + 0.309017i
503.1 −0.891007 + 0.453990i 0 0.587785 0.809017i −0.987688 0.156434i 0 0 −0.156434 + 0.987688i 0 0.951057 0.309017i
503.2 0.891007 0.453990i 0 0.587785 0.809017i 0.987688 + 0.156434i 0 0 0.156434 0.987688i 0 0.951057 0.309017i
647.1 −0.453990 0.891007i 0 −0.587785 + 0.809017i 0.156434 0.987688i 0 0 0.987688 + 0.156434i 0 −0.951057 + 0.309017i
647.2 0.453990 + 0.891007i 0 −0.587785 + 0.809017i −0.156434 + 0.987688i 0 0 −0.987688 0.156434i 0 −0.951057 + 0.309017i
683.1 −0.453990 + 0.891007i 0 −0.587785 0.809017i 0.156434 + 0.987688i 0 0 0.987688 0.156434i 0 −0.951057 0.309017i
683.2 0.453990 0.891007i 0 −0.587785 0.809017i −0.156434 0.987688i 0 0 −0.987688 + 0.156434i 0 −0.951057 0.309017i
827.1 −0.156434 + 0.987688i 0 −0.951057 0.309017i −0.891007 0.453990i 0 0 0.453990 0.891007i 0 0.587785 0.809017i
827.2 0.156434 0.987688i 0 −0.951057 0.309017i 0.891007 + 0.453990i 0 0 −0.453990 + 0.891007i 0 0.587785 0.809017i
863.1 −0.156434 0.987688i 0 −0.951057 + 0.309017i −0.891007 + 0.453990i 0 0 0.453990 + 0.891007i 0 0.587785 + 0.809017i
863.2 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 0.891007 0.453990i 0 0 −0.453990 0.891007i 0 0.587785 + 0.809017i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner
100.l even 20 1 inner
300.u odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.1.bh.a 16
3.b odd 2 1 inner 900.1.bh.a 16
4.b odd 2 1 CM 900.1.bh.a 16
12.b even 2 1 inner 900.1.bh.a 16
25.f odd 20 1 inner 900.1.bh.a 16
75.l even 20 1 inner 900.1.bh.a 16
100.l even 20 1 inner 900.1.bh.a 16
300.u odd 20 1 inner 900.1.bh.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.1.bh.a 16 1.a even 1 1 trivial
900.1.bh.a 16 3.b odd 2 1 inner
900.1.bh.a 16 4.b odd 2 1 CM
900.1.bh.a 16 12.b even 2 1 inner
900.1.bh.a 16 25.f odd 20 1 inner
900.1.bh.a 16 75.l even 20 1 inner
900.1.bh.a 16 100.l even 20 1 inner
900.1.bh.a 16 300.u odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( ( 1 - 6 T + 13 T^{2} - 10 T^{3} + 16 T^{4} - 10 T^{5} + 2 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$17$ \( 625 + 125 T^{4} + 150 T^{8} - 20 T^{12} + T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
$31$ \( T^{16} \)
$37$ \( ( 1 + 4 T - 2 T^{2} - 10 T^{3} + 16 T^{4} - 10 T^{5} + 7 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$41$ \( 1 - 16 T^{2} + 97 T^{4} + 32 T^{6} + 150 T^{8} - 32 T^{10} + 12 T^{12} - 4 T^{14} + T^{16} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( 625 - 500 T^{4} + 150 T^{8} + 5 T^{12} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( ( 25 + 25 T^{2} + 10 T^{4} + T^{8} )^{2} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( ( 1 + 6 T + 13 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( 1 - 4 T^{2} + 92 T^{4} + 228 T^{6} + 230 T^{8} + 72 T^{10} + 17 T^{12} + 4 T^{14} + T^{16} \)
$97$ \( ( 1 + 6 T + 13 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} )^{2} \)
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